To estimate the data-space inverse we need to approximate an
inverse for the cross product matrix *LL*^{T}.
This inverse acts as a filter
for the data space. Each element of measures the correlation between a data
element and another data element . The computation of each
element requires the evaluation of an inner product in the
model space.
Since the model space is regularly sampled,
the inner products can then be computed analytically (pending
available representation of the chained operator, ).

Considering an irregularly sampled input of *n* seismic traces,
the cross-product operator *LL*^{T} can be expressed in matrix notations as:

(49) |

Each inner product is a reconstruction of a data trace with offset to a new trace with offset . Therefore the mapping is an AMO transformation, and can be written as

(50) |

where is AMO from input offset to output offset and, is the identity operator (mapping from to ). Conforming
to the definition of AMO,
is the adjoint of ; therefore, the filter is Hermitian
with diagonal elements being the identity and off-diagonal elements being
trace to trace AMO transforms.
This is a fundamental definition of **D** that will allow
a fast and efficient numerical approximation of its inverse.

The data-space inverse can then be expressed as a two-step solution
where the data is
first filtered with the inverse of the operator **D** then the adjoint
is applied to the filtered data to solve for a model.
The solution for from equation equ3 can be written
in terms of **D** as:

(51) |

(52) |

(53) |

we need now to solve for by computing the inverse of from the system of equations:

(54) |

Once the inverse of is estimated to yield the filtered data , we merely solve for the initial model .

Notice that after filtering, we can apply any imaging operator to invert for .

1/18/2001